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Solve -3x² - 12x - 9 = 0

The solutions of -3x² - 12x - 9 = 0 are x = -3 or x = -1.

x² +x += 0
−3x2−12x−9=0⇒x=−3, −1-3x^2 - 12x - 9 = 0 \Rightarrow x = -3,\ -1
Discriminant D = 36 · x ≈ -3, -1

Change a, b or c (whole numbers, a not 0) and the exact solutions follow.

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Graph of y = -3x² - 12x - 9-5-4-3-2-11-35-30-25-20-15-10-55x = -3x = -1vertexy = -3x² - 12x - 9
Graph of y = -3x² - 12x - 9. The curve crosses the x-axis at x = -3 and x = -1; the vertex is at (-2, 3).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -3, b = -12, c = -9.

    −3x2−12x−9=0-3x^2 - 12x - 9 = 0
  2. 2

    Work out the discriminant D = b² − 4ac. Its sign tells you how many real solutions there are.

    D=(−12)2−4⋅(−3)⋅(−9)=144−108=36D = (-12)^2 - 4\cdot(-3)\cdot(-9) = 144 - 108 = 36
  3. 3

    D is positive, so there are two different real solutions.

  4. 4

    Method 1, factoring. The roots are rational, so the left side factors:

    −3(x+3)(x+1)=0-3\left(x + 3\right)\left(x + 1\right) = 0
  5. 5

    A product is zero when one of its factors is zero, so set each factor equal to 0 and solve:

    x+3=0⇒x=−3x+1=0⇒x=−1x + 3 = 0 \Rightarrow x = -3\qquad x + 1 = 0 \Rightarrow x = -1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=12±36−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{36}}{-6}
  7. 7

    Simplify to get the solutions:

    x=−3orx=−1x = -3\quad\text{or}\quad x = -1
  8. 8

    Check by putting x = -3 back in: a·x² + b·x + c = -3·(9) + (-12)·(-3) + (-9) = 0 ✓

LaTeX source of the formulas on this page
\[ -3x^2 - 12x - 9 = 0\ \Rightarrow\ x = -3,\ -1 \]
\[ -3x^2 - 12x - 9 = 0 \]
\[ D = (-12)^2 - 4\cdot(-3)\cdot(-9) = 144 - 108 = 36 \]
\[ -3\left(x + 3\right)\left(x + 1\right) = 0 \]
\[ x + 3 = 0 \Rightarrow x = -3\qquad x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{36}}{-6} \]
\[ x = -3\quad\text{or}\quad x = -1 \]

The parabola y = -3x² - 12x - 9

xy
-5-24
-4-9
-30
-23
-10
0-9
1-24

Questions

What are the solutions of -3x² - 12x - 9 = 0?

The solutions of -3x² - 12x - 9 = 0 are x = -3 or x = -1.

Does -3x² - 12x - 9 = 0 factor?

Yes: -3(x+3)(x+1) = 0.

What is the vertex of y = -3x² - 12x - 9?

(-2, 3). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -4, product = c/a = 3.

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